| L(s) = 1 | − 4·2-s + 8·4-s − 16·8-s + 36·16-s + 4·25-s − 4·29-s − 8·31-s − 64·32-s + 8·37-s + 12·41-s − 16·50-s + 16·58-s + 32·62-s + 96·64-s − 16·67-s − 32·74-s − 48·82-s − 44·83-s − 44·97-s + 32·100-s − 20·101-s − 4·103-s − 20·107-s − 32·116-s − 64·124-s + 127-s − 160·128-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 5.65·8-s + 9·16-s + 4/5·25-s − 0.742·29-s − 1.43·31-s − 11.3·32-s + 1.31·37-s + 1.87·41-s − 2.26·50-s + 2.10·58-s + 4.06·62-s + 12·64-s − 1.95·67-s − 3.71·74-s − 5.30·82-s − 4.82·83-s − 4.46·97-s + 16/5·100-s − 1.99·101-s − 0.394·103-s − 1.93·107-s − 2.97·116-s − 5.74·124-s + 0.0887·127-s − 14.1·128-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2557953709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2557953709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 + 71 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 1719 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 56 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 39 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 75 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 136 T^{2} + 8850 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 4914 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 4146 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 + 7199 T^{4} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 8 T + 123 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 17046 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 10583 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 280 T^{2} + 31839 T^{4} - 280 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 22 T + 284 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 11706 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 22 T + 303 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.18042316966116352973238955377, −6.91640644071799403432208224159, −6.85376579955310919489387166688, −6.64233513006178215115024219096, −6.08652269230692867887335725413, −6.03455344245560014102902830137, −5.76880046904930439047080089569, −5.64385405429922356882402156564, −5.53466305957202522583329757537, −5.26632154045005837401244074699, −4.78063941161779800669111880346, −4.34501893011440195652884527751, −4.19785855885297822798387553332, −4.09715820804017197080351248140, −3.61825974582625382618737559207, −3.32229103393495401983605506956, −2.91859571619178403073735613636, −2.83176142733592146565588436664, −2.51192184762693109167390480548, −2.49646597532610248424389764603, −1.51560713840839301525470472294, −1.51479700569388072458689185184, −1.32394844943069010665554476915, −0.49805166085674209123712188965, −0.36095503284257890623002008479,
0.36095503284257890623002008479, 0.49805166085674209123712188965, 1.32394844943069010665554476915, 1.51479700569388072458689185184, 1.51560713840839301525470472294, 2.49646597532610248424389764603, 2.51192184762693109167390480548, 2.83176142733592146565588436664, 2.91859571619178403073735613636, 3.32229103393495401983605506956, 3.61825974582625382618737559207, 4.09715820804017197080351248140, 4.19785855885297822798387553332, 4.34501893011440195652884527751, 4.78063941161779800669111880346, 5.26632154045005837401244074699, 5.53466305957202522583329757537, 5.64385405429922356882402156564, 5.76880046904930439047080089569, 6.03455344245560014102902830137, 6.08652269230692867887335725413, 6.64233513006178215115024219096, 6.85376579955310919489387166688, 6.91640644071799403432208224159, 7.18042316966116352973238955377
